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# Existence of solutions of n-point boundary value problems on the half-line in Banach spaces

Henan University of Science and Technology Department of Mathematics and Physics Luoyang 471003 People’s Republic of China

Acta Applicandae Mathematicae (Impact Factor: 0.99). 01/2010; 110(2):785-795. DOI: 10.1007/s10440-009-9475-8 - Citations (13)
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**ABSTRACT:**The author considers the boundary value problem (1)y '' =f(x,y,y ' ),0≤x<∞, (2)a 0 y(0)=a 1 y ' (0)=A, a 0 ≥0, a 1 ≥0, a 0 +a 1 >0, (3)y(∞)=B· The basic assumptions on the function f(x,y,z) are: f(x,y,z) is continuous on I×ℝ 2 , I=[a,b], for 0<b<∞; f(x,y,z) is nondecreasing in y for each fixed pair (x,z)∈I×ℝ; f(x,y,z) satisfies a uniform Lipschitz condition on each compact subset of I×ℝ 2 with respect to z; and zf(x,y,z)≤0 for (x,y,z)∈I×ℝ 2 , z≠0. Using the shooting method, and with additional assumptions on f(x,y,z) and supposing that a 0 , a 1 are both positive, he proves that the boundary value problem (1)-(3) has a unique solution. The following example y '' =-2xy ' /(1-αy) 1/2 ,0≤x<∞, y(0)=1,y(∞)=0, which arises in nonlinear mechanics in the problem of unsteady flow of gas through a semi-infinite porous medium, 0<α≤1, is given.Journal of Mathematical Analysis and Applications 03/1990; 147(1):122–133. · 1.05 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the existence of positive solutions of boundary value problems on the half-line for differential equations of second order. The Krasnoselskii fixed point theorem on cone compression and expansion is used.Journal of Mathematical Analysis and Applications 07/2001; · 1.05 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Let us consider the question of existence of a solution of the problem: Find T>0 and y∈C[0,T]∩C 1 [0,T)∩C 2 (0,T) such that (1)y '' (t)+q(t)y(t) γ =0,t∈(0,T),y(0)=y(T)=0,y ' (0)=α,y(t)>0,t∈(0,T), where γ≥1, α>0 and q, a positive function, are given. If q(t)=t β , β a real number, (1) reduces to the Emden- Fowler equation whose origin lies in theories concerning gaseous dynamics in astrophysics. Under the additional assumptions: i) q∈C 2 (0,+∞), ∀t>0: q(t)>0, t→t γ , q(t) belongs to L 1 (0,1)· ii)η,η ”∈L 1 (1,+∞), where η∈C 2 (0,+∞) is defined by η(t)=[q(t)] -1/γ+3 and ∫ 1 +∞ [η(t)] -2 dt=+∞· iii) If γ>1: lim t↓0 t -1 η(t)[∫ t +∞ η(s)|η '' (s)|ds] 2/γ-1 =0, the author proves that the problem (1.1) has a unique solution (T,y). If q(t)=t β , it is shown that (1) has a solution if and only if γ+2β+3>0. Moreover a monotone iteration scheme holds for the approximation of a positive solution.Reviewer: A.CañadaSiam Journal on Mathematical Analysis - SIAM J MATH ANAL. 01/1987; 18(2).

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